Dynamic Choice under Ambiguity

Transcription

1 Dynamic Choice under Ambiguity Marciano Siniscalchi October 28, 2010 Abstract This paper analyzes dynamic choice for decision makers whose preferences violate Savage s Sure-Thing principle [40], and therefore give rise to violations of dynamic consistency. The consistent-planning approach introduced by Strotz [46] provides one way to deal with dynamic inconsistencies; however, consistent planning is typically interpreted as a solution concept for a game played by multiple selves of the same individual. The main result of this paper shows that consistent planning under uncertainty is fully characterized by suitable behavioral assumptions on the individual s preferences over decision trees. In particular, knowledge of ex-ante preferences over trees is sufficient to characterize the behavior of a consistent planner. The results thus enable a fully decision-theoretic analysis of dynamic choice with dynamically inconsistent preferences. The analysis accommodates arbitrary decision models and updating rules; in particular, no restriction need be imposed on risk attitudes and sensitivity to ambiguity. 1 Introduction In a dynamic-choice problem under uncertainty, a decision maker (DM henceforth) acquires information gradually over time, and takes actions in multiple periods and information scenar- Economics Department, Northwestern University, Evanston, IL 60208; marciano@northwestern.edu. I have greatly benefited from many extensive conversations with Peter Klibanoff and Nabil al-najjar. I also thank the Coeditor, Bart Lipman, and two anonymous referees, as well as Pierpaolo Battigalli, Eddie Dekel, Larry Epstein, Paolo Ghirardato, Alessandro Lizzeri, Fabio Maccheroni, Alessandro Pavan and Larry Samuelson for their valuable insights, and Simone Galperti for excellent research assistance. All errors are my own. 1

2 ios. The basic formulation of expected utility (EU) theory instead concerns a reduced-form, atemporal environment, wherein preferences are defined over maps from a state space Ω to a set of prizes X ( acts ). Thus, in order to analyze dynamic choice problems, it is necessary to augment the atemporal EU theory with assumptions about the individual s preferences at different decision points. The standard assumption is of course Bayesian updating: if the individual s initial beliefs are characterized by the probability q, her beliefs at any subsequent decision point h are given by the conditional probability q( B), where the event B represents the information available to the individual at h. Together with the assumption that the individual s risk preferences do not change, Bayesian updating ensures that the DM s behavior satisfies a crucial property, dynamic consistency (DC): the course of action that the individual deems optimal at a given decision point h, on the basis of the preferences she holds at h, is also optimal when evaluated from the perspective of any earlier decision point h (and conversely, if h is reached with positive probability starting from h ). This implies in particular that backward induction or dynamic programming can be applied to identify optimal plans of action. Bayesian updating and the DC property are intimately related to the cornerstone of Savage s axiomatization of EU, namely his Postulate P2; Section 2 discusses this tight connection and provides references. Sensitivity to ambiguity (Ellsberg [7]), or to the common-ratio or common consequence effects (Allais [2]; Starmer [45]), and other manifestations of non-eu risk attitudes, typically lead to violations of Savage s Postulate P2. As a consequence, violations of DC are to be expected when such preferences are employed to analyze dynamic choice problems; again, Section 2 elaborates on this point, and provides illustrative examples. These violations of DC, and ways to address them, are the focus of the present paper. Whenever a conflict arises among preferences at different decision points, additional assumptions are required to make clear-cut behavioral predictions (Epstein and Schneider [9], p. 7). One approach, introduced by Strotz [46] in the context of deterministic choice with changing time preferences and tastes, is to assume that the DM adopts the strategy of Consistent Planning (CP). In Strotz s own words, at every decision point, a consistent planner chooses the best plan among those that [s]he will actually follow ([46], p. 173). Formally, CP is a refinement of backward induction that incorporates a specific tie-breaking rule. Informally, CP reflects the intuitive notion that the DM is sophisticated: that is, she holds 2

3 correct beliefs about her own future choices. The problem with this intuitive notion is that, of course, beliefs about future choices cannot be observed directly; they also cannot be elicited on the basis of the DM s initial and/or conditional preferences over acts. The literature on time-inconsistent preferences circumvents this difficulty by suggesting that CP is best viewed as a solution concept for a game played by multiple selves of the same individual. Strotz himself ([46, p. 179]) explicitly writes that [t]he individual over time is an infinity of individuals ; see also Karni and Safra [28, pp ]), O Donoghue and Rabin [36, p. 106], and Piccione and Rubinstein [37, p. 17]. However, at the very least, this interpretation represents a major departure from the standard economics conception of the individual as the unit of agency (Gul and Pesendorfer [21, p. 30]). It certainly does not clarify what it means for an individual decision-maker to adopt the strategy of consistent planning. It reinforces the perception that a sound, behavioral analysis of multi-period choice requires some form of dynamic consistency (Epstein and Schneider [9, p. 2]). Finally, it provides very little guidance as regards policy analysis. This paper addresses these issues by providing a fully behavioral analysis of CP in the context of dynamic choice under uncertainty. In the spirit of the menu-choice literature initiated by Kreps [31], I assume that the individual is characterized by a single, ex-ante preference relation over dynamic choice problems, modeled as decision trees. I then show that: under suitable assumptions, conditional preferences can be derived from ex-ante preferences over trees, regardless of whether or not preferences over acts satisfy Savage s postulate P2 (cf. Sec. 4.2 and Theorem 2); sophistication can be formalized as a behavioral axiom on preferences over trees, regardless of whether or not DC holds (cf. Sec ); and the proposed sophistication axiom, plus auxiliary assumptions, provides a behavioral rationale for CP (Theorems 3 and 4), again regardless of whether or not P2 or DC hold. Three features of the analysis in this paper deserve special emphasis. First, the approach in this paper is fully behavioral in the specific sense that the implications of CP are entirely reflected in the individual s ex-ante preferences over trees, which are observable. 3

4 Second, by providing a formal definition of sophistication that does not involve multiple selves, this paper provides a way to interpret this intuitive notion as a behavioral principle but one that applies to preferences over trees, rather than acts. The analysis also indicates that seemingly minor differences in the way sophistication is formalized can have significant consequences in the context of choice under uncertainty; see Sec Third, minimal assumptions are required on preferences over acts: the substantive requirements considered in this paper are imposed on preferences over trees. In particular, postulate P2 and hence DC play no role in the analysis. This allows for prior and conditional preferences that exhibit a broad range of attitude toward risk and ambiguity a main objective of the present paper. The main results in this paper do not restrict attention to any specific model of choice, or updating rule. However, to exemplify the approach taken here, Theorem 5 specializes Theorem 3 to the case of multiple-priors preferences (Gilboa and Schmeidler, [15]) and prior-byprior updating. Furthermore, Sec leverages the framework and results in this paper to address what is often cited as a paradoxical implication of CP (e.g. Machina [34], Epstein and Le Breton [8]): a time-inconsistent, but sophisticated DM may forego freely available information, if by doing so she also limits her future options. The analysis in shows that this behavior actually has a simple rationalization if preferences over trees, rather than just acts, are taken into account. Organization of the paper. Sec. 2 illustrates the key issues by means of examples. Sec. 3 introduces the required notation and terminology. Sec. 4 presents the main results, the special case of multiple-priors preferences, and the application to value-of-information problems. Sec. 5 discusses the main results. Sec. 6 discusses the important connections with the existing, rich literature on dynamic choice under ambiguity, as well as work on menu choice, intertemporal choice with changing tastes, and dynamic choice with non-expected utility preferences. 2 Heuristic treatment Savage s P2 and DC. It was asserted above that Bayesian updating and DC are intimately related to Savage s postulate P2; this implies that failures of DC are not pathological, but rather 4

5 the norm, when non-eu preferences are employed to analyze problems of choice under uncertainty. Savage himself provides an argument along these lines in [40, 2.7]; Ghirardato [13] formally establishes the equivalence of DC and Bayesian Updating with P2, under suitable ancillary assumptions. Proposition 1 in the present paper provides a corresponding, slightly more general equivalence result in the framework adopted here. 1 These results can be illustrated in simple examples that are also useful to describe the proposed behavioral approach to CP. An example. An urn contains 90 amber, blue and green balls; in the following, I shall consider different assumptions about what the DM knows regarding its composition. A single ball will be drawn; denote the corresponding state space by Ω = {α,β,γ}, in the obvious notation. At time 0, without knowing the prevailing state, the DM can choose a safe action s that yields a prize of 1 if the ball is amber or blue, and x {0, 1} otherwise; alternatively, the DM can choose 2 to place a contingent bet c. In this case, the DM receives x if the ball is green, and can place a bet on amber (a ) or blue (b ) at time 1 otherwise. The situation is depicted in Fig. 1: solid circles denote decision points, and empty circles denote points where Nature moves, or more properly reveals information to the DM. c s α,β γ γ x x a b α β α β α,β 1 2 Figure 1: A dynamic decision problem; x {0, 1}. Given the state space Ω and prize space X = {0, 1, 1}, the atemporal choice enviroment cor- 2 responding to the decision problem under consideration consists of all acts (functions) h X Ω. Suppose first that the DM knows the composition of the urn, and that she has risk-neutral EU preferences; her beliefs q (Ω) reflect the composition of the urn. Thus, in the atemporal 1 All versions of this argument incorporate the assumptions of consequentialism and (with the exception of Prop. 1 in this paper) reduction; the discussion of these substantive hypotheses is deferred till Sec

6 setting, the DM evaluates acts h = (h α, h β, h γ ) X Ω according to the functional V (h) = E q [h]. Now, as described above, augment this basic preference specification by assuming that the DM updates her beliefs q in the usual way. At the second decision node, she then conditionally (weakly) prefers a to b if and only if q({α} {α,β}) q({β} {α,β}). This is of course equivalent to q({α}) q({β}), which is the restriction on ex-ante belief that ensures that, from the point of view of the initial node, the course of action c then a is weakly preferred to c then b. This is an instance of DC: the ex-ante and conditional rankings of the actions a and b coincide. In turn, this provides a rationale for the use of backward induction: the plans of action available to the DM at the first decision node are c then a, c then b and s ; but one of the two c plans can be eliminated by first solving the choice problem at the second node. A simple calculation then shows that s is never strictly preferred, regardless of the ratio of blue vs. green balls. Hence, for instance, if q({a }) > q({b}), then c then a is the unique optimal plan. 2 To provide a concrete illustration of the relationship between DC and P2, recall that the assumptions of ex-ante EU preferences and Bayesian updating delivered two conclusions: (i) the ranking of a vs. b at the second decision is the same as the ranking of c then a vs. c then b at the first decision node; furthermore, (ii) the ranking of a vs. b at the second node is independent of the value of x. Now assume that the modeler does not know that ex-ante preferences conform to EU, nor that conditional preferences are derived by Bayesian updating; however, he does know that (i) and (ii) hold. Clearly, the modeler is still able to conclude that the ranking of c then a and c then b must also be independent of x, so that (1, 0, 0) (0, 1, 0) (1, 0, 1) (0, 1, 1), (1) where denotes the DM s preferences over acts. This is an implication of Savage s Postulate P2 (cf. [40, p. 23], or Axiom 4.2 in 4.1 below). In other words, as claimed, Eq. (1) is also a necessary condition for Dynamic Consistency in Fig As per footnote 1, this argument incorporates the substantive assumptions of Consequentialism and Reduction (see Sec. 3.3). In the tree of Fig. 1, the relevant aspect of Consequentialism is the fact that that the ranking of a vs. b at the second decision node is independent of the value of x ; Reduction instead implies that the choice of c followed by, say, a is evaluated by applying the functional V ( ) to the associated mapping from states to prizes, i.e. (1, 0,x ). I maintain both assumptions in this Introduction; the formal results in the body of the paper allow for arbitrary departures from Reduction. 6

7 Ambiguity, DC and CP. I now describe ambiguity-sensitive preferences that violate P2, and hence yield a failure of DC; see below for an analogous example based on the common-consequence effect. Assume that, as in the three-color-urn version of the Ellsberg paradox [7], the DM is only told that the urn contains 30 amber balls. Assume that she initially holds multiple-priors (a.k.a. maxmin-expected utility, or MEU) preferences (Gilboa and Schmeidler [15]), is risk-neutral for simplicity, and updates her beliefs prior-by-prior (e.g. Jaffray [26], Pires [38]) upon learning that the ball drawn is not green. Formally, her preferences over acts h Ω conditional on either F = Ω or F = {α,β}, are given by V F (h) = min q C E q [h F ], where C is the set of all probabilities q on Ω such that q({α}) = 1. Notice that such conditional preferences are independent of the 3 value of x, as is the case for Bayesian updates of EU preferences. Note first that, a priori (i.e. conditional on F = Ω), this DM exhibits the modal preferences reported by Ellsberg [7]: she prefers a bet on amber to a bet on blue, but she also prefers betting on blue or green rather than amber or green. Therefore, the DM s preferences violate Eq. (1), hence Savage s postulate P2. Furthermore, conditional on {α,β}, this DM prefers (1, 0,x) to (0, 1,x) regardless of the value of x, and hence will strictly prefer a to b. If now x = 1, DC is violated: at the first decision node, the DM strictly prefers the plan c followed by b to c followed by a, but at the second node she strictly prefers a to b. To resolve these inconsistencies, suppose that the DM adopts CP. The intuitive assumption of sophistication implies that, at the first decision node, the DM should correctly anticipate her future choice of a, regardless of the value of x. This is true despite the fact that, for x = 1, she really would like to commit to choosing b instead. Hence, when contemplating the choices c and s at the first decision node, the DM understand that she is really comparing the plan c then a to s. For x = 0, she will strictly prefer the former; but, for x = 1, she will strictly prefer the latter. This logic thus delivers unambiguous and coherent behavioral predictions. A dynamic common consequence paradox (cf. Allais [2]). Violations of DC can also arise when preferences are probabilistically sophisticated but not EU; again, CP provides a way to deal with them. Suppose that one ball will be drawn from an urn containing 100 balls, numbered 1 through 100. Fig. 2 depicts the choice problem and payoffs, where M denotes one million (dollars), and , 12..., 100 etc. refer to the number on the ball drawn. The DM s beliefs are uniform on Ω = {1,..., 100} at the initial node, and determined via 7

8 a M c b 1M s x 0.9M Figure 2: A dynamic Allais-type problem; x {0, 1M }. Bayes rule at the second; her preferences are of the rank-dependent EU form (Quiggin [39]), with quadratic distortion function. If x = 1M, the plan c followed by b is preferred to c followed by a, whereas the opposite holds if x = 0: this corresponds to the usual violations of the Independence axiom, and hence of P2. Furthermore, the DM strictly prefers a to b at the second decision node if x = 1M, so preferences are dynamically inconsistent. Nevertheless, CP again delivers well-defined behavioral predictions: if x = 1M, the DM will correctly anticipate choosing a at the second node, and hence, by a simple calculation, opt for s at the initial node. Karni and Safra [27, 28] illustrate applications of CP with non-eu preferences under risk. Behavioral analysis of CP. As noted in the Introduction, this paper provides a fully behavioral analysis of CP. To illustrate the key ingredients of the analysis, refer back to the decision tree in Fig. 1, and adopt a simplified version of the notation to be introduced in Sec. 3 (an analogous treatment can be provided for the tree in Fig. 2). Denote the original tree in Fig. 1 by f x ; also denote by c x and s x the subtrees of f x where c or, respectively, s is the only action available at the initial node. Finally, denote by ca x and cb x the subtrees of c x where a or, respectively, b is the only action available at the second decision node; note that s x, ca x and cb x can be interpreted as fully specified plans of action. Assume that, at time 0, the DM expresses the following strict preferences ( ) and indifferences ( ) over decision trees: ca 0 c 0 f 0 s 0 cb 0 and cb 1 f 1 s 1 ca 1 c 1. (2) The preferences in Eq. (2) exhibit two key features. First, preferences over plans are consistent with act preferences in the Ellsberg paradox, and more generally with the assumed MEU preferences at the initial node. Specifically, ca 0 s 0 cb 0 and ca 1 s 1 cb 1 correspond to the DM s 8

9 ranking of the acts (1, 0,x), (0, 1,x) and ( 1, 1,x ) for x = 0, 1 provided by the MEU utility index V 2 2 Ω. The remaining preference rankings involve non-degenerate trees, and do not merely follow from the assumption of MEU preferences (even if augmented with prior-by-prior updating); rather, they reflect the intuition behind sophistication that is the focus of this paper. In particular, the indifference c 1 ca 1 indicates that the DM does not value the option to choose b at time 1, when a is also available. This is not because she dislikes action b from the perspective of time 0: on the contrary, the ranking cb 1 ca 1 suggests that she would like to commit to choosing b at time 1. Therefore, it must be the case that this DM correctly anticipates her future strict preference for a over b, and evaluates the tree c 1 accordingly. I emphasize that this argument relies crucially upon the rankings of non-degenerate trees e.g., c 1 ca 1 in Eq. (2). Indeed, this pattern of preferences will constitute the behavioral definition of Sophistication in More generally, the proposed approach leverages preferences over trees to elicit conditional preferences and analyze sophistication and related behavioral traits, just like the literature on menu choice leverages preferences over menus to investigate attitudes toward flexibility or commitment, as well as temptation and self-control (see 6). 3 The preferences in Eq. (2) indicate how this particular DM resolves the conflict between her prior and posterior preferences. Furthermore, the rankings f 0 ca 0 and f 1 s may be intepreted as the behavioral implications of Sophistication: if x = 0, the DM will choose c and plan to follow with a, and if x = 1, she will choose s instead as predicted by CP. It was just argued that, if the DM is assumed to strictly prefer a to b at the second decision node, then the prior preferences in Eq. (2) reveal that she is sophisticated. But, reversing one s perspective, the following interpretation is equally legitimate: if the DM is assumed to be sophisticated, then the prior preferences in Eq. (2) reveal her ranking of a vs. b at the second decision node. To elaborate, as noted above, the rankings cb 1 ca 1 c 1 suggest that the DM expects to choose a rather than b at the second decision node; if the DM is assumed to be sophisticated, this expectation must be correct, so she must actually prefer a to b at that node. In this respect, the DM s prior preference relation over trees, partially described in Eq. (2), 3 Although I assumed that Reduction holds in this specific example, the notation and formal setup allow the DM to strictly rank two plans p, p that can be reduced to the same act. This is orthogonal to the issue of sophistication; imposing Reduction throughout would neither simplify nor hamper the analysis. See

10 provides all the information required to analyze behavior in this example. Details. Certain subtle aspects of CP in the context of choice under uncertainty require further analysis, and are fully dealt with in the remainder of this paper. First, eliciting conditional preferences in general trees requires a more refined approach than the one just described; the details are provided in Sec Note that only a weak form of sophistication is required. Second, ties must be handled with care. The Sophistication axiom in Sec is purposely formulated so as to entail no restrictions in case multiple optimal actions exist at a node. Instead, a separate axiom captures the tie-breaking assumption that characterizes CP. Third, this division of labor is essential in the setting of choice under uncertainty. Sec. 5.2 shows that, under solvability conditions that are satisfied by virtually all known parametric models of non-eu preferences, strengthening the Sophistication axiom so as to deal with ties as well has an undesirable side effect: it imposes a version of P2 on preferences over acts, and hence, for instance, rules out the modal preferences in the Ellsberg example. 3 Decision Setting Due to the approach taken in this paper, the notation for decision trees must serve two purposes. First, it must provide a rigorous description of dynamic-choice problem; second, it must allow a precise, yet relatively straightforward formalization of tree-surgery operations pruning actions at a given node, replacing actions at a node with different ones, and more generally composing new trees out of old ones. The proposed description of decision trees will be relatively familiar; 4 however, formally describing tree-surgery operations requires a level of detail that is not needed in other treatments of dynamic choice under uncertainty. For simplicity, attention is restricted to finite trees associated with a single, fixed sequence of information partitions; see 5.3 for possible extensions. 4 Epstein [11] and Epstein, Noor and Sandroni [12] adopt a similar notation for decision trees, although they are not motivated by (and do not define) tree-surgery operations. In the context of risk, the notation in Sec. 3 of Kreps and Porteus [32] is similar, again except for tree surgery; see Sec. 6 for further details. 10

11 3.1 Actions, Trees and Histories Fix a state space Ω, endowed with an algebra Σ, and a connected and separable space X of outcomes. Information is modeled as a sequence of progressively finer partitions 0,..., T of Ω, for some 0 T <, such that 0 = {Ω} and t Σ for all t = 1,..., T (sometimes referred to as a filtration). For every t = 0,..., T, the cell of the partition t containing the state ω Ω is denoted by t (ω); also, a pair (t,ω), where t {0,..., T } and ω Ω, will be referred to as a node. Trees and actions can now be defined recursively, as menus of contingent menus of contingent menus.... A bit more rigorously, define first a tree beginning at the terminal date T in state ω simply as an outcome x X. Inductively, define an action available in node (t,ω) as a map associating with each state ω t (ω) a continuation tree beginning at node (t + 1,ω ); to complete the inductive step, define a tree beginning at node (t,ω) as a finite collection, or menus, of actions available at (t,ω). The details are as follows: Definition 1 Let F T (ω) = F T = X for all ω Ω. Inductively, for t = T 1,..., 0 and ω Ω, let 1. A t (ω) be the set of t +1 -measurable functions a : t (ω) F t +1 such that, for all ω t +1 (ω), a (ω ) F t +1 (ω ); 2. F t (ω) be the collection of non-empty, finite subsets of A t (ω); and 3. F t = F ω Ω t (ω). The elements of A t (ω) and F t are called actions and trees respectively. Observe that the maps ω A t (ω) and ω F t (ω) are t measurable. A tree is interpreted throughout as an exhaustive description of the choices available in a given decision problem; in particular, if two or more actions are available at a node, the individual cannot also randomize among them. Of course, randomization can be explicitly modeled, by suitably extending the state space and the description of the tree. A history describes a possible path connecting two nodes in a tree: specifically, it indicates the actions taken and events observed along the path. Given the filtration 0,..., T, the sequence of events observed is fully determined by the prevailing state of nature; thus, formally, a history is identified by the initial time t, the prevailing state ω, and the (possibly empty) sequence of actions taken. The details, and some related notation and terminology, are as follows: 11

12 Definition 2 A history starting at a node (t,ω) is a tuple h = [t,ω, a], where either a = (a t,..., a τ ), with t τ T 1, a t A t (ω) and, for all t = t + 1,...,τ, a t a t 1(ω); or a = (an empty list). The cardinality of a is denoted a. Furthermore: 1. If h = [t,ω, a], a = and a t A t (ω), then a a t = (a t ); and if a = (a t,..., a τ ), τ < T 1 and a τ+1 a τ (ω), then a a τ+1 (a t,..., a τ, a τ+1 ). 2. A history [t,ω, a] is terminal iff t + a = T, and initial iff a =. 3. A history h = [t,ω, a] is consistent with a tree f F t (ω) if t = t, ω t (ω), and either a = or the first action in a is an element of f ; in this case, the continuation tree of f starting at h is f (h) = f if a =, and f (h) = a τ (ω ) if a = (a t,..., a τ ). Certain special trees play an important role in the analysis. First, a plan is a tree where a single action is available at every decision point. Formally, a tree f F t is a plan if, for every history h = [t,ω, a] consistent with f, f (h) = 1. The set of plans in F t and F t (ω) will be denoted by F p t and F p t (ω) respectively. Second, a constant plan yields the same outcome in every state of the world. Formally, f x t,ω F t (ω) is the unique plan such that, for every terminal history h consistent with f x, f x (h) = x. If the node (t,ω) can be understood from the context, the plan t,ω t,ω f x t,ω will be denoted simply by x. As an example, the tree in Fig. 1, as well as its subtrees, can be formally defined as follows (recall that a simplified notation was used in the Introduction). Let T = 2, 1 = {{α,β},{γ}}, and 2 = {{α},{β},{γ}}. The two choices available at the second decision node in Fig. 1 correspond to the time-1 actions a,b A 1 (α) = A 1 (β) defined by a (α) = 1, a (β) = 0 and b(α) = 0, b(β) = 1. (3) Next, define the time-0 actions c x, s x, ca x, cb x A 0 (α) = A 0 (β) = A 0 (γ) by for ω = α,β, c x (ω) = {a,b}, s x (ω) = 1 2, ca x (ω) = {a }, cb x (ω) = {b}; (4) c x (γ) = s x (γ) = ca x (γ) = cb x (γ) = x. (5) Here, x and 1 2 denote the constant plans f x 1,γ and f 1 2 1,γ respectively. 12

13 Now the full tree in Fig. 1 is formally defined as f x {c x, s x }; the subtree beginning with the choice of c (respectively, s ) is {c x } (respectively {s x }); and the plans corresponding to the choice of c at the initial node, followed by a (respectively b ) at the second decision node are {ca x } and {cb x }. Finally, there are three non-terminal histories consistent with f x :, [0,α, c x ] and [0,β,c x ]. 3.2 Composite Trees Fix f F t, a history h = [t,ω, a] consistent with f, and another tree g F t + a (ω). The composite tree g h f is, intuitively, a tree that coincides with f everywhere except at history h, where it coincides with g. Formalizing this notion is somewhat delicate, so I first provide some heuristics. Since h = [t,ω, a] is consistent with f and a = (a t,..., a τ ), with τ t, the last element a τ of the action list a satisfies a τ (ω ) = f (h) for all ω τ+1 (ω). To capture the idea that f (h) is replaced with g, one would like to replace a τ in the list a with a new action ā τ such that ā τ (ω ) = g at such states, and ā τ (ω ) = a τ (ω ) elsewhere. However, recall that, by definition, a τ 1 (ω ) must contain a τ for all ω τ (ω); if a τ is replaced with ā τ, it is also necessary to modify a τ 1 so that it now contains ā τ rather than a τ in such states. These modifications must be carried out inductively for all actions a τ 1, a τ 2,..., a t ; this yields a new, well-defined action list ā = (ā t,..., ā τ ). Finally, recall that, by definition, the history h = [t,ω, a] is consistent with f precisely when the first action a t in the list a is an element of f (trees are sets of actions). Then, the tree g h f differs from f precisely in that the action a t is replaced with ā t. Now for the formal details. If a =, then let g h f g. Otherwise, write a = (a t,..., a τ ), with τ t ; let ā τ (ω ) = g for all ω τ+1 (ω), and ā τ (ω ) = a τ (ω ) for ω τ (ω) \ τ+1 (ω). Inductively, for t = τ 1,..., t, let ā t (ω ) = {ā t +1} (a t (ω ) \ {a t +1}) for all ω t +1(ω), and ā t (ω ) = a t (ω ) for ω t (ω) \ t +1(ω). Finally, let g h f denote the set {ā t } (f \ {a t }). As a special case, consider a node (t,ω) and a plan f F p 0. Since, by definition, a single action is available in f at any node, there is a unique history consistent with f that corresponds to the node (t,ω); it is then possible to define a tree that, informally, coincides with f everywhere except at time t, in case event t (ω) occurs. Such tree will be denoted g t,ω f. Formally, since f is a t -period plan, there is a unique action list a = (a 0,..., a t 1 ) such that 13

14 h = [0,ω, a] is consistent with f. Then, for all g F t (ω), let g t,ω f g h f. 5 The notation g t,ω f is modeled after g E f, which is often used to indicate composite Savage acts. 3.3 Preferences, Reduction and Consequentialism Definition 3 A conditional preference system (CPS) is a tuple ( t,ω ) 0 t <T,ω Ω, such that, for every t and ω, t,ω is a binary relation on F t (ω), and furthermore ω t (ω) implies t,ω = t,ω. The time-0 preference is also denoted simply by. Three aspects are worth emphasizing. First, preferences are assumed to be adapted to : for each t = 0,..., T 1, t,ω is measurable with respect to t. This reflects the assumption that t,ω is the DM s ranking of trees conditional upon observing the event t (ω) at time t. Second, recall that, in the Introduction, preferences over plans were implicitly defined by first reducing plans to acts in the obvious way, and then invoking the DM s preferences over acts, represented by the functional V. While this is the textbook approach to dynamic choice with EU preferences, there are compelling reasons to consider alternatives. For instance, the DM may display a preference for early or late resolution of the uncertainty, as in Kreps and Porteus [32], Epstein and Zin [10], Segal [41] and, in a fully subjective setting, Klibanoff and Ozdenoren [30]. To allow for such preference models, reduction is not assumed in the main results of this paper, Theorems 2 and 3. The proposed approach takes the DM s preferences over plans as given, as part of her CPS, regardless of whether or not they are obtained from underlying preferences over acts by reduction. Third, the assumption that the only conditioning information relevant to the preference relation t,ω is the event t (ω) implies that our analysis is consequentialist: in particular, two actions a,b A t (ω) are ranked in the same way, in any decision tree where they may be available. To elaborate, if the actions a and b are available at a history h = [t,ω, a] consistent with a tree f, their ranking will of course, depend upon the realized event t (ω); however, prior choices made and alternatives discarded on the path to h, choices the DM would have had to make at counterfactual histories in f, or events that might have obtained but didn t, are irrele- 5 a is also the unique list such that, for any ω t (ω), the history h = [0,ω, a] is consistent with f ; furthermore, the definition of composite trees implies that g h f = g h f, consistently with the intended interpretation of g t,ω f. 14

15 vant. This is a standard property of EU preferences and Bayesian updating, and is preserved in most applications of non-eu and ambiguity-sensitive preferences. However, some alternative theoretical approaches to dynamic choice with non-eu preferences or under ambiguity relax consequentialism to salvage dynamic consistency; this important point is discussed in Sec. 6. To conclude, recall that a binary relation is a weak order iff it is complete and transitive. 4 Main Results This section presents the main results of this paper. Theorem 2 in 4.2 shows that sophistication provides a way to elicit conditional preferences over acts and trees from prior preferences over trees. 4.3 then takes as primitive a CPS and provides a definition (Def. 5) and characterization (Theorems 3 and 4) of CP in the context of choice under uncertainty considers CP for MEU preferences and prior-by-prior updating, and analyzes the value of information under CP. All proofs are in the Appendix. Further motivation and discussion is provided in 5. As a preliminary step, 4.1 formalizes the connection between dynamic consistency, Bayesian updating, and Savage s Postulate P2 mentioned in the Introduction. This result constitutes a useful benchmark, and aids in the interpretation of Theorems Dynamic Consistency, Bayesian updating, and Postulate P2 The main result of this subsection should be considered a folk theorem : various versions of it exist in the literature, beginning with Savage s own ([40, 2.7]). Its original statement concerns preferences over acts; I restate it in terms of preferences over plans merely to avoid introducing new notation. 6 Also note that, while the definition of a CPS involves general trees, throughout this subsection axioms, definitions and results are explicitly restricted to preferences over plans. For simplicity, assume that every event in the filtration 0,..., T is not Savage-null: for every node (t,ω), it is not the case that p r t,ω p for all p F p 0 and r F p t. I begin by formalizing DC, Savage s Postulate P2, and Savage s qualitative notion of Bayesian Updating; I follow Savage [40], 2.7 throughout, to which the reader is referred for interpretation (for DC, see also Epstein 6 That is, to further clarify, the resulting additional generality is inessential for my purposes. 15

16 and Schneider [9]). Note however that Savage s postulates and definitions pertain to all possible conditioning events, whereas I restrict attention to the elements of the filtration 0,..., T. Axiom 4.1 (Dynamic Consistency DC) For all nodes (t,ω) with t < T, and all actions a,b A t (ω) such that {a },{b} F p t : if a (ω ) t +1,ω b(ω ) for all ω t (ω), then {a } t,ω {b}; furthermore, if time-(t + 1) preferences are strict for some ω t (ω), then {a } t,ω {b}. 7 Axiom 4.2 (Postulate P2) For all plans p,q F p 0 r t,ω p s t,ω p r t,ω q s t,ω q. 8 p, all nodes (t,ω), and all plans r, s Ft (ω), Note that DC relates preferences at different histories; on the other hand, P2 pertains to prior preferences alone. As asserted in Sec. 2, the MEU preferences specified in Sec. 2, jointly with the reduction assumption, yield a violation of Axiom 4.2: take r = {a }, s = {b}, p = {ca 0 } and q = {cb 0 }. This is, of course, the main message conveyed by Ellsberg [7]. Finally, say that the restriction of t,ω to F p t Savage [40], p. 22) if, for all plans r, s F p t, is derived from via Bayesian updating (cf. r t,ω s r t,ω p s t,ω p for some plan p F 0. For ex-ante EU preferences the above condition indeed characterizes Bayesian updating of the DM s prior. The following result is then straightforward: 9 Proposition 1 Consider a CPS ( t,ω ) t,ω. The following statements are equivalent: (1) is a weak order on F p 0, Axiom 4.2 (Postulate P2) holds, and for every node (t,ω), the restriction of t,ω to F p t is derived from via Bayesian updating; (2) every t,ω is a weak order on F p t, and Axiom 4.1 (DC) holds. This result depends upon the assumption of consequentialism implicit in the framework: according to Def. 5, each preference t,ω is defined on (sub)trees with initial event t (ω) (cf. 6). For a version of this result that relaxes consequentialism, see Epstein and Le Breton [8]. 7 a and b are actions, whereas the singleton sets {a } and {b} are trees; on the other hand, a (ω ) and b(ω ) are trees in F t +1 (ω). Finally, F p t is a set of plans, i.e. special types of trees, and t,ω and t +1,ω are defined over trees. 8 A plan is, a fortiori, a t -period plan, so r t,ω p, etc. are well-defined: cf The proof is similar to that of analogous results (e.g. Ghirardato [13]); hence, it is available upon request. 16

17 Prop. 1 highlights the tension between dynamic consistency and ambiguity that was anticipated in the Introduction. However, I now wish to emphasize the implications of this result for Bayesian updating. If one assumes that prior preferences satisfy P2, then one can define conditional preferences via Bayesian updating, and in this case Prop. 1 implies that DC will hold. Conversely, if one assumes that DC holds, Prop. 1 implies that Bayesian updating provides a way to elicit conditional preferences; furthermore, ex-ante preferences necessarily satisfy P Eliciting Conditional Preferences Turn now to the main results of the paper, beginning with the elicitation of conditional preferences. First, we adopt a standard requirement: the conditioning event should matter. Assumption 4.1 (Non-null conditioning events) For every node (t, ω) and prizes x, y such that x y, there exists a plan g F p 0 such that x t,ωg y t,ω g. For general preferences over acts or plans, Assumption 4.1 is stronger than the requirement that every set t (ω) not be Savage-null (cf. 4.1); however, the two notions coincide, for instance, for MEU (and of course EU) preferences. Assumption 4.1 is weaker than analogous conditions in the literature e.g. the notions of non-null event in Ghirardato and Marinacci [14], which requires that g = y Beliefs about Conditional Preferences I begin by proposing a procedure that elicits the DM s beliefs about her own future preferences; the details are in Def. 4. To motivate it, refer to the decision tree in Fig. 1 with x = 1; adopt the notation in Eqs. (3) (5). Since {cb 1 } {ca 1 } {c 1 } ex-ante, it was argued in 2 that {a } 1,α {b} [equivalently, {a } 1,β {b}]: if the DM would like to commit to b at the second decision node, but deems the tree {c 1 } just as good as committing to a, it must be the case that the DM expects to choose a at the second decision node, if both a and b are available. However, this argument fails if {ca 1 } {cb 1 }: in this case, the indifference {c 1 } {ca 1 } is not sufficiently informative as to the relative conditional ranking of a vs. b. Detecting conditional indifferences is even more delicate. Thus, a different, but related approach must be adopted: 17

18 Definition 4 For all nodes (t,ω) and trees f, f F t (ω), f is conjecturally weakly preferred to f given (t,ω), written f 0 t,ω f, iff there exists a prize z X such that, for all plans g F p 0, y X, y z (f y ) t,ω g y t,ω g and z y (f y ) t,ω g f t,ω g. (6) The superscript 0 in the notation 0 emphasizes that this conjectural conditional preference t,ω relation is defined solely in terms of the DM s time-0, i.e. prior preferences. The logic behind Def. 4 is as follows. Suppose that the DM believes that f t,ω f. Under suitable regularity (in particular, solvability) assumptions that are captured by the axioms in the next subsection, there exists a prize z X such that (the DM will also believe that) f t,ω z t,ω f. Now consider another prize y X such that a priori y z ; if the DM does not expect her preferences over prizes to change, then (she will believe that) y t,ω z as well, and hence that y t,ω f. But this implies that she will expect y to be chosen rather than f in the tree (f y ) t,ω g at node (t,ω). 10 As in the example of Sec. 2, the ex-ante indifference between (f y ) t,ω g and y t,ω g now reflects this belief. The argument for the case z y is similar. Note that, for every tree considered in Eq. (6), there is a unique path from the initial history to the node (t,ω), because g is a plan; furthermore, the event t (ω) is not Savage-null. Hence, the DM cannot avoid contemplating her choices at that node Axioms and Characterization The axioms I consider are divided into two groups. Axioms relate the DM s actual conditional preferences with her prior preferences; Axioms instead concern the DM s prior preferences only, and ensure that the definition of conjectural conditional preferences (in Def. 4) is well-posed (that is, non-contradictory). Axiom 4.3 (Stable Tastes) For all x,x X, and all nodes (t,ω): x t,ω x if and only if x x. Axiom 4.4 (Conditional Dominance) For all nodes (t,ω), all f F t (ω), and all x,x X : if x f (h) x for all terminal histories h of f, then x t,ω f t,ω x. 10 f y denotes the time-t tree that contains all actions in f, plus the unique initial action in the plan f y t,ω that leads to the prize y in every state of nature. In other words, the notation exploits (a) the simplified notation for prizes, and (b) the fact that trees are just sets of acts, and therefore unions of trees are also well-defined trees. 18

19 Axiom 4.5 (Conditional Prize-Tree Continuity) For all nodes (t,ω) and all f F t (ω), the sets {x X : x t,ω f } and {x X : x t,ω f } are closed in X. Axiom 4.6 (Weak Sophistication) For all nodes (t,ω), plans g F p 0, trees f F t (ω), and prizes x X : x t,ω f (f x ) t,ω g x t,ω g and x t,ω f (f x) t,ω g f t,ω g. Axiom 4.3 states that tastes, i.e. preferences over prizes, are unaffected by conditioning. 11 Axioms 4.4 and 4.5 are standard, and ensure that conditional certainty equivalents exist (recall that X is assumed to be a connected and separable topological space). Axiom 4.6 assumes just enough sophistication to ensure that conjectural and actual conditional preferences coincide: in particular, the logic of sophistication is applied only to comparisons between a tree and a constant prize, and then only if the DM has no other choice available on the path to the node (t,ω). Preferences at times t > 0 are not required to be sophisticated. Turn now to the second group of axioms. Axiom 4.7 (Prize Continuity) For all x X, the sets {x X : x x} and {x X : x x} are closed in X. Axiom 4.8 (Dominance) Fix a node (t,ω), a tree f F t (ω), a plan g F p 0 (i) If f (h) x for all terminal histories h of f, then (f x) t,ω g f t,ω g. and a prize x X. (ii) If f (h) x for all terminal histories h of f, then (f x) t,ω g x t,ω g. Axiom 4.8 reflects stability of preferences over outcomes. If the individual s preferences over X do not change when conditioning on t (ω), then in (i) she will expect not to choose x at node (t,ω), because f yields strictly better outcomes at every terminal history; similarly for (ii). As in Sec. 2, the indifferences in (i) and (ii) capture the DM s expectations. The next axiom is a beliefs-based counterpart to Weak Sophistication: 11 For the present purposes, it would be sufficient to impose this requirement on a suitably rich subset of prizes. For instance, if X consists of consumption streams, it would be enough to restrict Axiom 4.3 to constant streams. 19

20 Axiom 4.9 (Separability) Consider a node (t,ω), f F t (ω), plans g, g F p 0 and x, y X. Then: (i) (f y ) t,ω g f t,ω g and x y imply (f x) t,ω g x t,ω g ; (ii) (f y ) t,ω g y t,ω g and x y imply (f x ) t,ω g f t,ω g. To interpret, consider first the case g = g and fix a prize y. According to the by-now familiar logic of belief elicitation, (f y ) t,ω g f t,ω g indicates that the DM believes that she will not strictly prefer f to y given t (ω) otherwise indifference would have to obtain. Thus, if x y and the DM s preferences over X are stable, she will also expect to strictly prefer x to f given t (ω); again, the elicitation logic yields (f x) t,ω g x t,ω g. The interpretation of (ii) is similar. Additionally, Axiom 4.9 implies that these conclusions are independent of the particular t -period plan under consideration, and hence of what the decision problem looks like if the event t (ω) does not obtain. In this respect, Axiom 4.9 reflects a form of separability. More generally, Axiom 4.9 essentially requires that Eq. (6) in Def. 4 hold for all plans g, or for none. There is a close analogy with the role of Savage s Postulate P2: see 4.1 for details. The main result of this section can now be stated. Theorem 2 Suppose that Assumption 4.1 holds. Consider the CPS ( t,ω ), and assume that is a weak order on F 0. Then the following statements are equivalent. 1. satisfies Axioms ; furthermore, for all nodes (t,ω), t,ω = 0 t,ω. 2. For every node (t,ω), t,ω is a weak order, and satisfies Axioms Theorem 2 and Proposition 1 in 4.1 are structurally similar: Axioms play the role of Postulate P2 (but add solvability requirements), the definition of conjectural conditional preferences corresponds to Bayesian updating, and Axioms correspond to DC (but again add solvability requirements). The interpretation is also similar: under Axioms , Def. 4 yields well-behaved conditional preferences, and hence can be taken as the definition of conditional preferences; in this case, Axioms will hold. Conversely, if Axioms hold, the beliefs derived via Def. 4 from prior preferences are actually correct, so that Def. 4 can be seen as a way to elicit actual conditional preferences. The main differences are that, of course, Theorem 2 does not rely on P2 or DC, and concerns preferences over non-degenerate trees. 20

21 4.3 A decision-theoretic analysis of Consistent Planning Consistent Planning under Uncertainty As noted in the Introduction, Consistent Planning (CP) is a refinement of backward induction. If there are unique optimal actions at any point in the tree, the two concepts coincide. Otherwise, CP complements backward induction with a specific tie-breaking rule: indifferences at a history h are resolved by considering preferences at the history that immediately precedes h. To illustrate, consider the tree in Fig. 1 with x = 1, but assume MEU preferences with priors C = {q (Ω) : 1 90 q(α) 30 90, q(β) }. Continue to assume prior-by-prior updating and 90 reduction, and again adopt the notation in Eqs. (3) (5). It can then be verified that {a } 1,α {b}; however, {ca 1 } {cb 1 }, so CP prescribes that the DM will follow c with a. The corresponding plan {ca 1 } is strictly preferred to {s 1 }, so the unique CP solution of this tree is the plan {ca 1 }. Algorithmically, CP operates as follows. For each history h = [t,ω, a] in a tree f, consider first the set CP 0 f (h) of actions b A t + a (ω) that, for every realization ω t + a (ω), prescribe a continuation action a t + a +1,ω that has survived prior iterations of the procedure. Intuitively, such actions b correspond to plans that the DM will actually follow. Then, out of these actions, select the conditionally optimal ones: this completes the induction step and defines the set CP f (h). Def. 5 is modeled after analogous definitions in Strotz [46] and Gul and Pesendorfer [20], except that it is phrased in terms of preferences, rather than numerical representations. Definition 5 (Consistent Planning) Consider a tree f F t (ω). For every terminal history h = [t,ω, a] consistent with f, let CP f (h) = {f (h)}. Inductively, if h = [t,ω, a] is consistent with f and CP f ([t,ω, a a ]) has been defined for every ω t + a (ω) and a f (h), let CP 0 (h) = A f t + a (ω) : a f (h) s.t. ω t + a (ω), b b(ω ) = {a +1,ω } for some a +1,ω CP f ([t,ω, a a ]) CP f (h) = b CP 0 (h) : a f CP0 (h), {b} f t + a,ω {a }. and A plan {a } F t (ω) is a consistent-planning solution of f if a CP f ([t,ω, ]) To help parse notation, a, a +1,ω and b in this definition are acts; b(ω ) must therefore be a tree, and in particular the definition requires that it be the tree {a +1,ω } having a single initial action a +1,ω taken from the set CP f ([t,ω, a a ]). Finally, braces in {b} t + a,ω {a } are required because t + a,ω is defined over trees, not actions. 21

22 Note that, in order to carry out the CP procedure, it is only necessary to specify the DM s preferences over plans. The output of the CP algorithm is also a set of plans. 13 Moreover, it is straightforward to verify that, if preferences over plans are complete and transitive, then Def. 5 is wellposed: it always delivers a non-empty set of solutions that the DM deems equally good Behavioral Characterization of Consistent Planning The behavioral analysis of CP takes as input the DM s CPS ( t,ω ). The key assumption of Sophistication was introduced in Sec. 2; Axiom 4.6 applies the same principle to a small set of trees, with unique features. To capture the implications of Sophistication in general trees, it will be assumed that pruning conditionally dominated actions leaves the DM indifferent. Formally, if g is a subset of actions available in the tree f at the history h, and every action b g is strictly preferred to every action w that lies in f (h) but not in g, then ex-ante the DM must be indifferent between f and the tree g h f in which the inferior actions have been pruned: Axiom 4.10 (Sophistication) For all f F t, all histories h = [t,ω, a] consistent with f and such that a, and all g f (h): if, for all b g and w f (h) \ g, {b} t + a,ω {w }, then f t,ω g h f. Observe that Axiom 4.10 is silent as far as indifferences at node (t + a,ω) are concerned. For instance, if f (h) = {a,b} and {a } t + a,ω,{b}, the axiom does not require that f t,ω {a } h f t,ω {b} h f. This allows for the possibility that, ex-ante, the DM have a strict preference for commitment to a or b ; Axiom 4.11 deals with these situations. Axiom 4.10 is also silent if h is the initial history of f : Axiom 4.12 below encodes the assumptions required in this case. This division of labor is crucial so as to avoid unduly restricting ambiguity attitudes: see 5.2. The next axiom formalizes the tie-breaking assumption that characterizes CP within the class of backward-induction solutions: if the DM is indifferent among two or more actions at a history h, then she can precommit (more precisely, expects to be able to precommit) to any of them at the history that immediately precedes h. It is important to emphasize that no such precommitment is possible in case the individual has strict preferences over actions at h: in such cases, the full force of the Sophistication axiom applies. 13 Formally, CP f ([t,ω, ]) is a set of actions, not plans; however, if a CP f ([t,ω, ]), then {a } is a plan. 22